Copulas have been used extensively to capture cross-sectional dependence
in a multivariate stochastic
process \(\lbrace \mathbf {Y}_t\rbrace \) , where \(\mathbf {Y}_t=(Y_{1,t},\ldots ,Y_{d,t})^\top \) ; see [1]}, [2]}, [3]}
and [4]} for just some examples.
These models typically capture serial dependence through existing
marginal time series models; for example,
heteroscedastic models are normally used for financial returns.
An alternative is to use a single high-dimensional—but parsimonious—copula to
capture both serial and cross-sectional dependence jointly.
An advantage of this approach is that the marginal
distribution of each variable can be modeled directly, including
as non-parametric. It is this type
of time series copula that is the focus of this section.
